$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$ $\newcommand{\bfnu}{\boldsymbol{\nu}}$

Problems to Chapter 10

1. Problem 1
2. Problem 2
3. Problem 3
4. Problem 4
5. Problem 5

Problems

There are several classical problems

Problem 1. The heavy flexible but unstretchable wire (chain) has a length and an energy respectively \begin{gather} L= \int_0^a \sqrt{1+u^{\prime\,2}}\,dx, \label{eq-10.P.1}\\ U=\rho g \int_0^a u \sqrt{1+u^{\prime\,2}}\,dx \label{eq-10.P.2} \end{gather} where $\rho$ is a linear density.

1. Write down an equation minimizing energy $U$ as length $L$ is fixed.
2. Find solution satisfying $u(0)=h_0$, $u(a)=h_1$.

Problem 2. We need to construct the fastest slide from point $(0,0)$ to $(a,-h)$. If $u(x)$ describes its shape then time is \begin{equation} T= \int_0^a \frac{1}{\sqrt{2gu}} \sqrt{1+u^{\prime\,2}}\,dx. \label{eq-10.P.3} \end{equation}

1. Write down an equation minimizing energy $U$ as length $L$ is fixed.
2. Find solution satisfying $u(0)=0$, $u(a)=-h$.

Problem 3. If in 2D–light propagation the speed of light at point $(x,y)$ is $c(x,y)$ then the time of travel between two points $(x_1,y_1)$ and $(x_2,y_2)$ equals \begin{equation} T=\int_{x_1}^{x_2} \frac{\sqrt{1+u'^2}\,dx}{c(x,u(x))} \label{eq-10.P.4} \end{equation} where $y=u(x)$ is equation of the ray, $u(x_j)=y_j$, $j=1,2$.

1. Write down Euler' equation.
2. Consider the case of $c(x,y)$ depending only on $y$ and reduce the 1-order equation to 1-st order one using conservation law $H(u,u')=\const$ where $H=u' L_{u'}-L$ is a corresponding Hamiltonian, $L$ is a Lagrangian.

Problem 4. The area of the surface is \begin{equation} S=\iint_{D} \sqrt{1+u_x^2+u_y^2}\,dxdy \label{eq-10.P.5} \end{equation} where $z=u(x,y)$, $(x,y)\in D$ is an equation of the surface.

1. Write Euler-Lagrange PDE of the surface of the minimal area (with boundary conditions $u(x,y)=\phi(x,y)$ as $(x,y)\in \Gamma$ which is the boundary of $D$).
2. if the potential energy is \begin{equation} E= kS - \iint_D fu\,dxdy \label{eq-10.P.6} \end{equation} with $S$ defined by (\ref{eq-10.P.5}) and $f$ areal density of external force.

Write Euler-Lagrange PDE of the surface of the minimal energy.

Problem 5.

If the surface is a surface of revolution $z=u(r )$ with $r^2=x^2+y^2$ then \begin{equation} S=2\pi\int_{D} \sqrt{1+u_r^2}\,rdr \label{eq-10.P.7} \end{equation} Write Euler-Lagrange equation and solve it.

$\Leftarrow$  $\Uparrow$  $\Rightarrow$